# Sharpening the Becker-Stark Inequalities

## Abstract

In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.

## 1. Introduction

Steckin  (or see Mitrinovic [2, 3.4.19, page 246]) gives us a result as follows.

Theorem 1.1 (see [1, Lemma ]).

If , then (11)

Later, Becker and Stark  (or see Kuang [4, 5.1.102, page 248]) obtain the following two-sided rational approximation for .

Theorem 1.2.

Let , then (12)

Furthermore, and are the best constants in (1.2).

In fact, we can obtain the following further results.

Theorem 1.3.

Let , then (13)

Furthermore, and are the best constants in (1.3).

In this paper, in the form of (1.2) and (1.3) we shall show a general refinement of the Becker-Stark inequalities as follows.

Theorem 1.4.

Let , and let be a natural number. Then (14)

holds, where , and (15)

where are the even-indexed Bernoulli numbers.

Furthermore, and are the best constants in (1.4).

## 2. Four Lemmas

Lemma 2.1.

The function is decreasing, where is Riemann's zeta function.

Proof. is equivalent to the function , which is decreasing.

Lemma 2.2 (see [5, Theorem ]).

Let be Riemann's zeta function and the even-indexed Bernoulli numbers. Then (21)

Lemma 2.3 (see [6, 1.3.1.4 (1.3)]).

Let . Then (22)

Lemma 2.4.

Let and . Then , where (23)

Proof.

By Lemma 2.3, we have (24)

Since is decreasing by Lemma 2.1, it follows that (25)

From Lemma 2.2, we get (26)

which implies that for .

## 3. Proofs of Theorems

Proof of Theorem 1.4.

Let (31)

Then (32)

By Lemma 2.4, we have for , and is decreasing on .

At the same time, = by (3.1), and by (3.2), so and are the best constants in (1.4).

Proof of Theorem 1.3.

Let in Theorem 1.4; we obtain that and . Then the proof of Theorem 1.3 is complete.

## References

1. 1.

Steckin SB: Some remarks on trigonometric polynomials. Uspekhi Matematicheskikh Nauk 1955, 10(1(63)):159–166.

2. 2.

Mitrinovic DS: Analytic Inequalities. Springer, New York, NY, USA; 1970:xii+400.

3. 3.

Becker M, Strak EL: On a hierarchy of quolynomial inequalities for tanx. University of Beograd Publikacije Elektrotehnicki Fakultet. Serija Matematika i fizika 1978, (602–633):133–138.

4. 4.

Kuang JC: Applied Inequalities. 3rd edition. Shangdong Science and Technology Press, Jinan City, China; 2004.

5. 5.

Scharlau W, Opolka H: From Fermat to Minkowski, Undergraduate Texts in Mathematics. Springer, New York, NY, USA; 1985:xi+184.

6. 6.

Jeffrey A: Handbook of Mathematical Formulas and Integrals. 3rd edition. Elsevier Academic Press, San Diego, Calif, USA; 2004:xxvi+453.

## Author information

Authors

### Corresponding author

Correspondence to Ling Zhu.

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Zhu, L., Hua, J. Sharpening the Becker-Stark Inequalities. J Inequal Appl 2010, 931275 (2010). https://doi.org/10.1155/2010/931275 